From 5999e8682785cc397e266122fba91fafa8b48269 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 20 Feb 2021 14:55:33 +0100 Subject: Add "Dirac notation" + tweak "Bloch's theorem" --- static/know/concept/blochs-theorem/index.html | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'static/know/concept/blochs-theorem') diff --git a/static/know/concept/blochs-theorem/index.html b/static/know/concept/blochs-theorem/index.html index 6e5767c..f977739 100644 --- a/static/know/concept/blochs-theorem/index.html +++ b/static/know/concept/blochs-theorem/index.html @@ -59,7 +59,7 @@ \end{aligned} \]

In other words, in a periodic potential, the solutions are simply plane waves with a periodic modulation, known as Bloch functions or Bloch states.

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This is suprisingly easy to prove: if the Hamiltonian \(\hat{H}\) is lattice-periodic, then it will commute with the unitary translation operator \(\hat{T}(\vec{a})\), i.e. \(\comm{\hat{H}}{\hat{T}(\vec{a})} = 0\). Therefore \(\hat{H}\) and \(\hat{T}(\vec{a})\) must share eigenstates \(\psi(\vec{r})\):

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This is suprisingly easy to prove: if the Hamiltonian \(\hat{H}\) is lattice-periodic, then it will commute with the unitary translation operator \(\hat{T}(\vec{a})\), i.e. \([\hat{H}, \hat{T}(\vec{a})] = 0\). Therefore \(\hat{H}\) and \(\hat{T}(\vec{a})\) must share eigenstates \(\psi(\vec{r})\):

\[ \begin{aligned} \hat{H} \:\psi(\vec{r}) = E \:\psi(\vec{r}) -- cgit v1.2.3